Lesson 3: 7.3 The Sampling Distribution of a Sample Proportion

Why does the standard deviation of a proportion get smaller as the sample size n increases?

What are the "safety checks" (conditions) we must verify before we can use a Normal curve to find probabilities for a proportion?

How do we interpret the standard deviation of p-hat in a sentence?

Sample proportion (p-hat)

Population proportion (p)

Mean of the sampling distribution of p-hat

Standard deviation of the sampling distribution of p-hat

10% Condition (Independence)

Large Counts Condition (Normality)

HSS-ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.

HSS-IC.B.4: Use data from a sample survey to estimate a population proportion.

SAT questions often present a sample proportion and a margin of error. Understanding that the "true" p is likely within a few standard deviations of p-hat is the foundation for interpreting those results. If the SAT asks about the "confidence" or "likelihood" of a result, it is essentially asking about the area under the Normal curve for a sampling distribution.

This section provides the mathematical framework for analyzing "Yes/No" data. Students learn that if they take a large enough sample, the "pile" of possible sample proportions will form a beautiful Normal bell curve centered at the true population proportion.

DOK 2: Calculate the standard deviation of p-hat for a sample of 100 students if the true proportion of students who drive to school is 0.30.

DOK 3: Explain why the 10% condition is necessary for the standard deviation formula to be accurate when sampling without replacement.

The Problem: According to a recent report, 70% of high school students in a certain state have a part-time job (p = 0.70). You take a random sample of n = 150 students.

Task 1: Check if the conditions for using a Normal distribution are met.

Task 2: Find the probability that the sample proportion p-hat is less than 0.65.

Condition Confusion: Students often mix up the "10% Condition" (used to justify the standard deviation formula) and the "Large Counts Condition" (used to justify the Normal shape).

p-hat vs. p

Support: Provide a "Condition Placemat"—a laminated sheet where students must check off "Random," "10%," and "Large Counts" before they are allowed to touch their calculators.

Extension: Ask students to explore what happens to the standard deviation when p = 0.5 versus p = 0.9.

Teacher assigns examples from textbook and other resources.

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